Mathematicians Come Closer to Solving Goldbach's Weak Conjecture
scientificamerican.com
scientificamerican.com
http://terrytao.wordpress.com/
is not to be missed. The "Career Advice" section of the blog
http://terrytao.wordpress.com/career-advice/
includes several classic articles, of which I especially like and recommend "Does one have to be a genius to do maths?"
http://terrytao.wordpress.com/career-advice/does-one-have-to...
(Tao's answer is no.)
Terence Tao is a humble, self-effacing guy. Or at least he knows that society expects him to act like that rather than mimic the chest-thumping, endzone-dancing wide receiver. So obviously he's not going to say that "yeah, to really move math forward you kind of need to be a genius". Though of course other geniuses like Feynman were more up front about that truth.
Very few people have the innate ability to do what Tao or Feynman did. Recognizing that early is a mercy that allows people to move into other areas where they have a comparative advantage, rather than beating their head against the wall for 10,000 hours only not to get anywhere.
If you ever played a puzzle game (or dwarffortress), you should understand that "beating our head against the wall only not to get anywhere" could even be enjoyable. If you embark in this math adventure, maybe you will not become the next vonNeumann, but you will still have fun.
If you enjoy it, do it.
Terry Tao was clearly brilliant at mathematics as a child, though what nobody knew then was that he'd go on to be super-brilliant at mathematics as an adult, rather than being one of the many disappointing prodigies.
Anyway, I'm pretty smart, but I don't take career advice from anyone as smart as Terry Tao, just as I don't take dating advice from Scarlett Johannsen.
School mathematics isn't even math, it's a bad joke. I recommend Lockhart's Lament[1], I fully share his views. School 'math' destroyed every ounce of interest I had in math, and now, as a CS student, I still have a hard time getting rid of this attitude towards the subject even thought I know a lot better now.
If you're a Tao or Feynman, chances are you have been doing "maths" intuitively since a very young age. All these moments add up and reinforce each other. Is that the same as being a genius? Maybe, but then we are dealing with definitions - in my mind genius is a myth created by society to explain "unexplainable" things. That and an excuse for people's relative incompetence - see Hamming in his article "You and Your Research" [0].
The science of experts and deliberate practise [1, 2] is actually quite solid, despite its gladwellification. For example there have to my knowledge not been found a single person who defies the "logic of practise" - oft cited examples are Mozart and Woods, which are more of a myth making than based in any known facts (consider how both their fathers pushed them extremely hard with the right type of practise from a very early age).
Of course, there will always be variations (but these could in my opinion just as well be ascribed to right-time-right-place mechanisms). I suspect a large "problem" is - if the next 11 year old Tao has already had thousands of hours of something akin to deliberate practise, how on earth are other people - who don't share this natural inclination - going to catch up?
0: From http://www.cs.virginia.edu/~robins/YouAndYourResearch.html
Now for the matter of drive. You observe that most great scientists have tremendous drive. I worked for ten years with John Tukey at Bell Labs. He had tremendous drive. One day about three or four years after I joined, I discovered that John Tukey was slightly younger than I was. John was a genius and I clearly was not. Well I went storming into Bode's office and said, ``How can anybody my age know as much as John Tukey does?'' He leaned back in his chair, put his hands behind his head, grinned slightly, and said, ``You would be surprised Hamming, how much you would know if you worked as hard as he did that many years.'' I simply slunk out of the office!
What Bode was saying was this: ``Knowledge and productivity are like compound interest.'' Given two people of approximately the same ability and one person who works ten percent more than the other, the latter will more than twice outproduce the former. The more you know, the more you learn; the more you learn, the more you can do; the more you can do, the more the opportunity - it is very much like compound interest. I don't want to give you a rate, but it is a very high rate. Given two people with exactly the same ability, the one person who manages day in and day out to get in one more hour of thinking will be tremendously more productive over a lifetime. I took Bode's remark to heart; I spent a good deal more of my time for some years trying to work a bit harder and I found, in fact, I could get more work done. I don't like to say it in front of my wife, but I did sort of neglect her sometimes; I needed to study. You have to neglect things if you intend to get what you want done. There's no question about this.
On this matter of drive Edison says, ``Genius is 99% perspiration and 1% inspiration.'' He may have been exaggerating, but the idea is that solid work, steadily applied, gets you surprisingly far. The steady application of effort with a little bit more work, intelligently applied is what does it. That's the trouble; drive, misapplied, doesn't get you anywhere. I've often wondered why so many of my good friends at Bell Labs who worked as hard or harder than I did, didn't have so much to show for it. The misapplication of effort is a very serious matter. Just hard work is not enough - it must be applied sensibly.
1: http://en.wikipedia.org/wiki/Deliberate_practice#Deliberate_...
2: http://www.amazon.com/Cambridge-Expertise-Performance-Handbo...
It sounds like you are under the impression that success in mathematical olympiads is connected with success in mathematical research. It’s actually relatively rare for IMO winners to go on to become successful researchers, because the skills involved, while not completely disjoint, are not exactly the same either.
I don't see how when he started dating is relevant here...
On a serious note, I find this puzzle far more interesting than Fermat's Last Thereom, even though it doesn't seem to be well known at all. Probably because it hasn't been solved and subsequently been the subject of TV documentaries, admittedly. Are there any other comparably simple/intuitive rules in mathematics that have never been fully proven?
As mentioned in the article, it was proved over 50 years ago that every odd N can be written as a sum of three primes, provided N is sufficiently large. The proof extends to cover five primes instead, if you like (indeed, write N = 2 + 2 + (N - 4), but in fact the five primes case is in fact easier).
Usually in such proofs the definition of "sufficiently large" is maybe like 10^(10^(10^(10^(10^10000)))) or something similarly absurd, possibly far worse, and it is typically difficult even to calculate such a fixed N, because then you can't use big-O estimates in your calculations. Sometimes you can't even compute such an N, it is "ineffective", see e.g. here for an example:
http://en.wikipedia.org/wiki/Siegel_zero
Tao's accomplishment, which was really cool, was to bring the value of N down to the level where you could settle the small N case by brute force.
I don't think that makes sense. Every large odd number has exactly 0 ways to find its sum with two primes^, so they can't have more ways than small numbers (which also have 0, of course). Perhaps he was speaking about the strong conjecture, but then there would be no point to say 'let alone three', because three is only relevant for the weak conjecture.
Am I missing something obvious, or did the author just word that sentence/paragraph poorly?
^I am pretty sure this trivially follows from the fact that two odd numbers always produce an even number.
Though that doesn't directly imply the conjecture being true for sufficiently large numbers, since primes become increasingly sparse as one moves up the number line... it seems more like an appeal to intuition than a real argument, to me.